N95. 14095

SIMULATION OF A SPATIAL SERVO-HYDRAULIC TEST FACILITY FOR SPACE STRUCTURES

K.-D. Leimbach, H. Hahn

Control Engineering and System Theory Group,Department of Mechanical Engineering,

University of Kassel, Germany

ABSTRACT

In this paper different control concepts for servo-hydraulic test facilities are derived using exact linearizationtechniques. Based on different linear and nonlinear models of the test table and the actuator dynamics several nonlinear

controllers of different complexity are derived. The closed loop system performance of the controlled servo-hydraulictest facility is tested in various computer simulations using both, standard test signals and large test signals as system

inputs. The simulation results turn out, that in case of standard input signals the test facility controller must include alinear test table mechanics model and a nonlinear servohydraulic actuator model. Additional simulations demonstrate the

robustness of the control concept selected for standard test signals with respect to variations of plant parameters.

INTRODUCTION (1)

Multi-axis servohydraulic test facilities are widely used for vibration testing of critical components of industrialequipment and of future spacecraft. First theoretical investigations of multi-axis test facility control concepts based onmodern nonlinear control theory (exact linearization techniques) are (ref. 1) and (ref. 2). The control algorithms derived

in (ref. 1, 2) are based on complex nonlinear component models of the test table mechanics and on linear componentmodels of the hydraulic actuators. This paper extends the mathematical models of the servohydraulic actuators in (ref. 1,

2) from linear equations to highly nonlinear relations. Based on the different combinations of linear and nonlinear plantcomponent models (Section 2) different control algorithms have been derived (Section 3). Using the different control

algorithms of Section 3 various computer simulations have been made (Section 4) to find a trade-off between thecontroller complexity needed and the closed loop system behaviour desired. Additional computer simulations proof the

robustness of the control concept selected with respect to plant parameter variations. The control algorithm selectedprovides an excellent system performance and turns out to be amazingly insensitive with respect to plant parametervariations.

NONLINEAR PLANT MODELING (2)

The test facility considered includes the following components (cf. 1):

a rigid six degree of freedom test table with a rigid payload rigidly attached to it,/=6 or/=8 servohydraulic actuators and

an integrated control system.

Test table and payload mechanics

The nonlinear model equations of the test table mechanics are (ref. 1)

_, = r(x,).x2M(J,r_e , x,)'._2 =Jxr (x,)'A,'P-Jxr (xl)'D,'Jx](xl)'x2-n(J,r_e ,x 1 ,x2)--qG(rLp ,xt)

(la)

where the first equation of (la) includes the Poisson equations and the matrix T(xt) has been chosen as matrix of Bryant

angles; M is the inertia matrix of the test facility, D x and A x are the diagonal matrices of the actuator damping

coefficients and of the actuator piston areas, respectively; x I =[(ten) r , rlr]r=[x_, y_, z_, tp, O, _]rE_6 is the position

427

PAC,_ Bt.ANt( fKlqr lt11.M_

R Rand orientation vector of the spatial test facility, where the position vector re =[x e , yg, z_,] r is represented in inertial

frame R and the orientations r/=[cp, 0, _]r are the Bryant angles; x2 =[ffeR) r , (_)]r =[kee , 5_eR , _eR , p L, q z, r L]r _/R6 is

the velocity vector, where ¢0_ =[pL, qL, rt.] r is the angular velocity vector represented in body fixed frame L; k 2 is the

acceleration vector; n is the vector of the centrifugal forces and of the gyroscopic terms; q6 is the nonlinear vector of

the gravitational force and the associated torque and P:=[Pt,'", Pl] r e/RI is the vector of the actuator pressure

differences. The nonlinear transformation matrix Jrt(x 0 maps the actuator forces from joint fixed frames K_ into forces

and torques represented in inertial frame R and in body fixed frame L, respectively (compare the representation of x2).

The associated linear model equations are

M(J,r_p).)i I + Tf .Dx.Ta.£I=Tf "Ar'p

where Tf is the linearized transformation matrixcommon center of gravity.

(lb)

JxTl, JE_ 3'3 is the inertia tensor and rctee_ 3 is the vector of the

Servohydraulic actuators

The servohydraulic actuators are modeled by the nonlinear equations (ref. 3)

p= B2(P, Xv).xu-CHl.Qsp(p)-Cff.ax.Jx,(xl)'x2+Bl(p, xv)'Xv elR t (2a)

where (for i=1 ..... l, (/=6 in case of 6 actuators and/=8 in case 8 actuators acting onto the test table))

Bl(p,xv):=cz'.K,.{e,(e,Xv)+e,,(p,xv)}(p,Xv):=c;:.Ko.{Q,,(p,Xv)-Q,(p,Xv)}

(3)

and

Ks ::diag(otDi.lr.di. 2ff_) _t,t

QI :=diag( 4lpo - pi 112.sign(po - Pi ).a(xvi - xui )) _ _ "1

QII :=diag( 41Po+p,I/2.sign(Po+ Pi ).o'(-xvi - Xui )) _-IRt,t

Cn :=_.B.Ar (L actuator length, B bulk modulus).

The associated linear model equations are

(2b)

where Xv is the vector of servovalve piston positions Xv =[X_l,'", x_t] r E IRt and Cn , Qe, Q_ _lR_'t are the diagonal

matrices of the actuator hydraulic capacities, servovalve pressure coefficients and servovalve displacement coefficients,

respectively. The plant simulation includes the following model equations of the servovalve mechanics

2 2J/_i+2.(_i-¢,o_i .Jc_i+toni .x_i =kvi.aJ_i .u t elR 1, (i=l ..... 1) (4a)

with _'_ =0.7 and fv=O2v/27r. In (4a) u=[u_,..., ut]r_lR t is the system input. The control laws will be derived taking

into account the following simplified servovalve model

Xvi=kvi.ui E/R 1, (i=l ..... 1). (4b)

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NONLINEAR CONTROLLER DESIGN (3)

The subsequent nonlinear controller design is based on the trivial servovalve dynamics of (4b), where the servovalvedynamics included in (4a) has been neglected (ref. 1). This tremendously simplifies the controller design. Thissimplification will be justified by the subsequent simulation results. Then the choice of the system output y=h(xl):=x lleads to a well defined vector relative degree of [3,3,3,3,3,3] (in case of/=6 servo-hydraulic actuators). Using inaddition the system output y=x I yields the normal form representation (5) of the simplified plant which provides thebasis for the subsequent controller designs.

Zl :----._:J£1 :Z2

Z2 :=y=Jfl =Z 3

2 3 :---_aj -t-fj 'U ,

(5)

where _j =z2, Z2 =Z3 and Z3 are the spatial test table velocity, acceleration and jerk, respectively, and ctj, fj (j= I, II, III,IV) are the different normal form plant relations (7a), (7b), (7c), (7d) associated to the plant models collected in Table l,respectively.

Based on the system equations (5) the nonlinear controller design is straight forward (ref. 1):

u=f;t.(v-%) (6a)

where for l=6 fl_-I is the inverse of flj, and for l=8 fl._-i is defined as the right inverse of fj.

A suitable choice of K0, K1, K 2 E_ 6'6 in

V::Zsd--K2'(Zs--Zsd)--K1 "(Z2--Z2d)--K0"(Zl--Zld) (6b)

guarantees a stable error dynamics of the closed loop system, where Z3d' Z3d' Z2d' Zld are the desired jerk, acceleration,velocity and position set point signals, respectively. The measurements required in (6) are the test table degree offreedom positions/orientations zl, velocities z2 and accelerations z3. The plant nonlinearities are compensated by aj and

fjd (compare (6a)) where, the different representations of the controller relations % and fj_ depend on the plant com-ponent model equations of Table 1. The different controllers investigated are (comp. Table 1):

Case j=I (using (lb) and (2b)):

a, =M -1 .Tf .A K .{CH 1 .Qp'p-CH 1 "A x 'Ta'.iq }-M -1 "Tf "Dx'Ta'J_,

fit =M -I .Ta r .A r .CH 1 .Qp. K v,

(7a)

Case j=II (using (lb) and (2a)):

at, = M -1 .Tf .Ax .{B2.x,, -C_' "QnP -CH 1"At "Td"Jq}-M-I .Tf .O x .Ta .2 I

flu = M-' .Tf .A x .C H'.Ka "{Qt +Qtt }"Kv,(7b)

Case j=IlI (using (la) and (2b)):

d3xl(p,x,,x2) [o_ 1 .

d,3tim = T. M-I .Tf .A x .C_ I .Q,,,.K v,

Case j=IV (using (la) and (2a)):

d3Xl( p'xl ' x2)=V_____.b + °21/1 j: °aXl k 1a/V= dii L_I" O'IX'Ii-' I + _2-2' 2J

BIN = T'M-I .Tr.Ax.Ch 1"Ka{Qt +Qn }'Kv

(7d)

429

where

£1=-_-T(T(xl)'x2 )..i, + T(xl).{M-I(xl).[Jrxl(xl).Ag.p-jr (xl).DK.Jx,(xl).x2-n(x, , x2)-qa(xl)]}. (8)

Both, the controller complexity and the closed loop system behaviour severely depend on the plant models included in

the control law design. Control laws based on nonlinear test table equations of motion (la) (Case III and Case IV)are extreme complex and lengthy (more than one hundred DIN A4 pages of length, ref. 1). This is due to the partialderivatives of _ with respect to p, x_ and x 2 in (7c) and (7d), where i_ is defined in (8). Control laws based on linear

test table mechanics (1 b) are much less complex (less than 5 pages of length (Cases I and II)). The simulation results

of Section 4 show which component models must be included in the control law design in order to achieve a satisfactorycontrol loop behaviour using different test signal levels.

SIMULATION RESULTS (4)

Variation of control concepts (test facility with lffi6 servo-hydraulic actuators)

The subsequent simulations tend to find a trade-off between the closed loop system behaviour and the complexity ofthe controller to be implemented. Apart from Simulation No. 5b and 5c (comp. Table 2) all simulations are based on theideal servovalve model (4a) with an upper limit frequency of f, =1000 Hz. As discussed in Section 3 the controllercomplexity severely depends on the plant model equations taken in[o account in the controller design. The nonlinear plant(la), (2a) and (4a) together with the different control laws of Table 1 have been investigated in computer simulationsusing both, standard transient test signals for vibration testing of space structures used in industry (as shown in Figures2a and 4a) and large test signals (as shown in Figures 3a and 5a) derived from the standard test signals in order tosimulate large spatial motions. A control law based on linear plant models of both, the test table mechanics and theactuator dynamics (comp. Table 1, Case I and Equation (7a)) provides in case of standard test signals (cf. 2a) largecouplings among the test facility degrees of freedom and a poor tracking behaviour in the y- and z-degrees of freedom(cf. 2b and Simulation No. la of Table 2). A control law based on linear actuator dynamics and on nonlinear testtable mechanics (comp. Table 1, Case III and Equation (7c)) provides in case of standard test signals the same poortracking behaviour in the y- and z-degrees of freedom and large couplings among the test facility degrees of freedom (cf.2c and Simulation No. lb of Table 2). Even less quality results hold for simulations using large test signals (cf. 3a) incase of a control concept based on linear plant models (cf. 3b and Simulation No. 2a of Table 2) and in case of acontrol concept based on linear actuator dynamics and on nonlinear test table mechanics (cf. 3c and SimulationNo. 2b of Table 2). The unwanted couplings among all test table degrees of freedom may destroy the payloads to betested. Both, the tracking and the decoupling behaviour of the test facility have been improved by taking into accountlinear test table mechanics and nonlinear actuator dynamics in the control law design (comp. Table 1, Case IIand Equation (7b)) using standard test signals (cf. 4b and Simulation No. 3a of Table 2). A control concept based onnonlinear test table mechanics and on nonlinear actuator dynamics (comp. Table 1, Case IV and Equation (7d))provides the same excellent tracking and decoupling behaviour using standard test signals (cf. 4c and Simulation No. 3bof Table 2). In case of large test signals the control concept based on linear test table mechanics and on nonlinearactuator dynamics provides both, a poor tracking behaviour in the y- and z-degrees of freedom and large couplingsamong all degrees of freedom (cf. 5b and Simulation No. 4a of Table 2). A control concept based on nonlinear testtable mechanics and on nonlinear actuator dynamics provides an ideal tracking and decoupling behaviour even incase of large test signals (cf. 5c and Simulation No. 4b of table 2). From the previous computer simulations the followingconclusions have been drawn:

Using standard test signals (cf. 2a) a controller design based on linear test table mechanics and on nonlinearhydraulic actuator models provides a satisfactory transient and decoupling behaviour of the controlled test facility.

In case of large test signals (cf. 3a) both, the nonlinear test table mechanics and the nonlinear actuatordynamics must be included in the controller design. This, however, generates controllers of extreme high complexity(comp. Table 1, Case IV).

Additional computer simulation results with unstructured uncertainties are shown in Figure 6. The plantsimulation still includes the nonlinear test table mechanics (1 a), the nonlinear servo-hydraulic actuator dynamics (2a) andthe servovalve mechanics model (4a). The control concept selected is based on linear test table mechanics (lb), onnonlinear actuator dynamics (2a) and on the simplified servovalve model (4b) which doesn't take into account anyservovalve dynamics. In the simulations the upper servovalve limit frequency f of the plant model is varied fromfv=lO0 nz to fv =1000 Hz. In case of an unrealistic upper limit frequency of fv =1_0 Hz the transmission behaviour of

430

the test facility is nearly ideal (cf. 6a). As a consequence the valve dynamics (4a) can be omitted from the controller

design in this case. For fy =300 Hz the transmission behaviour is of reduced quality but still acceptable (cf. 6b). Forf =100 Hz the transmission behaviour is further reduced in comparison to Figure 6b and no longer acceptable (cf. 6c).OVn the other hand, Figure 6b shows, that an exact linearization controller design for a test facility including highresponse valves with an upper limit frequency fv of about 300 Hz may be drastically simplified by omitting theservovalve dynamics in the control law design. The resulting control errors are of minor practical importance.

Robustness with respect to plant parmneter variations (test facility with 1=8 servo-hydraulic actuators)

In industrial practice usually not all plant parameters to be implemented in the control algorithms are known exactly.As a consequence, the plant parameter values used in the control law may deviate from the actual parameter values of theplant. The influence of those parametric discrepancies between associated plant and controller parameters is investigatedsubsequently by the second set of computer simulations (comp. Table 3). The control concept selected for theseinvestigations is based on linear test table mechanics (lb), on nonlinear servohydraulic actuator dynamics(2a) and on the servovalve model (4b). Standard transient test signals used in industry for vibration testing of space

structures are used throughout these investigations and the valve limit frequency of the plant is set to fv =1000 Hz. Eachof the following variations of a controller parameter is counted with respect to the associated nominal value of thiscontroller parameter which is identical to the associated plant parameter. Variations of the servovalve amplificationfactor k,i by 10 % in Figure 7b produce only slight modifications in the tracking behaviour of the system. Onlyvariations of k_, of about 50 % or more have significant influence on the tracking behaviour (cf. 7c). The decoupling ofthe test table degrees of freedom is insensitive even with respect to large variations of k_. A variation of kv_simultaneously stands for a variation of various other controller amplifications factors. Variations of the bulkmodulus B which stand for variations of the compressibility due to temperature variations and due to oil contaminationscaused by air bubbles are shown in Figure 8. The tracking behaviour is only influenced slightly by variations of B of50% (cf. 8b). Unrealistic variations of B by a factor four or more modify the tracking behaviour significantly accordingto Figure 8c. In Figure 9b the common mass m of the test table and payload is modified by a factor 1/2. Thisdrastic mass variation again does not severely influence both, the tracking and the coupling behaviour of the system. Onlyunrealistic variations of m by a factor four or more have significant influence on the system behaviour (cf. 9c).Variations of all moments and products of inertia by a factor 1/2 or 4 have no significant effect on the systembehaviour (cf. 10b and cf. 10c). Variations of the common center of gravity r_e of the test facility and payload areshown in Figure ! 1. Simultaneous huge modifications of all components of the vector rcLe from reference point P tocenter of gravity C (for fixed P) by a factor of 1/2 are shown in Figure I lb. They don't affect the tracking behaviourand only provide slight couplings among the test table degrees of freedom. Only unrealistic variations of r_e by a factorfour or more show significant modifications of the transmission behaviour of the system (cf. 1 lc). The nominal plantmodel equations (5) did not include nonlinear friction forces due to dry friction in the hydraulic actuators and in thejoints attached to those. As a consequence those friction forces have not been taken into account in the nonlinear controllaw design. Figure 12 shows simulation results including nonlinear friction forces of different absolute values in the plantmodel. Friction forces FR smaller than 1 kN don't provide relevant influences upon the system behaviour (cf. 12b).Friction forces of the order of Fn =3 kN don't affect the tracking behaviour. They introduce coupling effects of some testfacility degrees of freedom (cf. 12c). Extreme high friction forces in the servo-hydraulic actuators are avoided inpractice by using actuators with hydrostatic bearings. The above numerical sensitivity analysis of the control systemdemonstrates that the exact linearization approach is much less sensitive to plant parameter variations and to controllerparameter variations than often predicted.

CONCLUSION (5)

This paper presents different control designs for spatial multi-axis test facilities with six or eight servo-hydraulicactuators. The various control designs investigated are based on exact linearization techniques taking into account

different combinations of linear and nonlinear plant component models. The system performance of the multi-axis testfacility controlled by different control concepts has been tested in various computer simulations. In case of standardtransient input signals a nonlinear controller design based on nonlinear servo-hydraulic actuator models and on linear test

table mechanics is sufficient to achieve an excellent tracking and decoupling behaviour. The amount of the associatedcontroller implementation is kept within acceptable limits. In case of large spatial transient motions of the test table withhigh velocities and accelerations, the control concept must take into account both, the nonlinear actuator dynamics and the

nonlinear test table mechanics. This yields control algorithms of several hundred pages of length which require anenormous amount of hardware implementation work. Additional simulation results demonstrate considerable robustness

of the closed loop system with respect to plant parameter variations and with respect to unmodeled servovalve dynamicsin the controller.

431

REFERENCES (6)

1. Hahn, H., Leimbach, K.-D.:. Nonlinear Control and Sensitivity Analysis of a Spatial Multi-Axis Servohydraulic TestFacility. Proceedings of the 32nd IEEE CDC, San Antonio 1993.

2. Hahn, H., Leimbach, K.-D. and Zhang, X.: Nonlinear Control of a Spatial Multi-Axis Servo-Hydraulic Test Facility.IFAC World Congress, Sydney 1993.

3. Hahn, H.: Theoretische Modellbildung Elektro-Servohydraulischer Antriebe, Lecture Notes, IMAT-Report 2,University of Kassel (ISSN 0939-8678), 1991.

Table 1. Plant models included in the controller design.

Test

SignalLevel

models of mechanical componentsincluded in the control law design

linear (lb)

Case I

(0_I) _11

_ (low controller_ "- complexity)0 mL)

,,_ ,_,

" .9 _ Case II

'_ (low controller=o complexity)

nonlinear (1 a)

Case lII

(O_IlI, _III)

(high controllercomplexity)

Case IV

(_rv, 13w)

(extreme highcontroller

complexity)

Table 2. Computer simulation runs of the controlled servo-hydraulic test facility (with 6 servo-

hydraulic actuators) using different control algorithms. The control algorithms are based ondifferent plant component models and nominal test facility model data.

SimulationNo.

la)Ib)

2a)

2b)

3a)3b)

4a)

4b)

5a)

5b)

5c)

Test signal

standard(Figure 2a)

large(Figure 3a)

standard(Figure 4a)

large(Figure 5a)

standard

(Figure 4a)

Case I

lin mech / lin hydFigure 2b

Control concepts

Case II

lin mech / nl hyd

Case III

nl mech / lin hyd

Figure 2cFigure 3b

Figure 4b

Figure 5b

Figure 6afv =1000 HzFigure 6b

fv=300 HzFigure 6c

fv =100 Hz

Figure 3c

Case IV

nl mech / nl h_,d

Figure 4c

Figure 5c

Grading of theclosed loop system

performance

tracking Idecouplin_

++ ++++ ++

++ ++

++ ++

++ ++

+

lin mech (nl mech) : linear (nonlinear) model of the test table mechanicslin hyd (nl hyd) : linear (nonlinear) model of the servo-hydraulic actuators

++: excellent tracking / decoupling behaviour+ : good tracking / decoupling behaviour- : poor tracking / decoupling behaviour

- -: very poor tracking / decoupling behaviour

432

Table 3. Computer simulation runs of the controlled servo-hydraulic test facility (with 8 servo-

hydraulic actuators) for plant parameter variations using the control law (7b) and standard

test signals.

Simulation!

No.

6a)

6b)

6c)

7a)

7b)

7c)

8a)

8b)

8c)

9a)

9b)

9c)

10a)

lOb)

iOc)

1la)

llb)

1lc)

lin mech :

nl hyd :

Test signal

standard

(Figure 2a)

Control concept

Case II

lin mech / nl hyd

Variation

servo valve gainfactor

kv

bulk modulus

B

common mass of

test table and pay-load m

inertia tensor

J

center of gravity oftest table and pay-

load r_e

dry friction forcewithin the actuators

PR

linear model of the test table mechanics

nonlinear model of the servo-hydraulic actuators

Grading of the

Variation Figure closed loop systemfactor performance

tracking [decoupling

1 (nominal value) 7a + + + +

0.9 7b + + +

1.5 7c + +

1 (nominal value) 8a + + + +

0.5 8b + +

4 8c +

! (nominal value) 9a + + + +

0._ 91;, + +

4 _ ....1 (nominal value) 10a + + + +

0.5 10b + + + +

4 i_ ++ ++1 (nominal value) I la + + + +

0.5 Ilb ++ +4 llc --

F_=0 kN 12a ++ ++

Fe=l kN 12b + + +

F/_=3 kN l_f + +

++ : excellent tracking / decoupling behaviour

+ : good tracking / decoupling behaviour

- : poor tracking / decoupling behaviour- - : very poor tracking / decoupling behaviour

Figure 1: Computer drawing of a multi-axis servo-hydraulic test facility.

433

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